Computational Particle Mechanics

, Volume 6, Issue 1, pp 11–28 | Cite as

Hysteretic behavior using the explicit material point method

  • Christos D. Sofianos
  • Vlasis K. Koumousis


The material point method (MPM) is an advancement of particle in cell method, in which Lagrangian bodies are discretized by a number of material points that hold all the properties and the state of the material. All internal variables, stress, strain, velocity, etc., which specify the current state, and are required to advance the solution, are stored in the material points. A background grid is employed to solve the governing equations by interpolating the material point data to the grid. The derived momentum conservation equations are solved at the grid nodes and information is transferred back to the material points and the background grid is reset, ready to handle the next iteration. In this work, the standard explicit MPM is extended to account for smooth elastoplastic material behavior with mixed isotropic and kinematic hardening and stiffness and strength degradation. The strains are decomposed into an elastic and an inelastic part according to the strain decomposition rule. To account for the different phases during elastic loading or unloading and smoothening the transition from the elastic to inelastic regime, two Heaviside-type functions are introduced. These act as switches and incorporate the yield function and the hardening laws to control the whole cyclic behavior. A single expression is thus established for the plastic multiplier for the whole range of stresses. This overpasses the need for a piecewise approach and a demanding bookkeeping mechanism especially when multilinear models are concerned that account for stiffness and strength degradation. The final form of the constitutive stress rate–strain rate relation incorporates the tangent modulus of elasticity, which now includes the Heaviside functions and gathers all the governing behavior, facilitating considerably the simulation of nonlinear response in the MPM framework. Numerical results are presented that validate the proposed formulation in the context of the MPM in comparison with finite element method and experimental results.


Material point method Plasticity Hysteresis Stiffness and strength degradation 



The authors would like to acknowledge the support from the “RESEARCH PROJECTS FOR EXCELLENCE IKY/SIEMENS”.


  1. 1.
    Andersen S, Andersen L (2010) Analysis of spatial interpolation in the material-point method. Comput Struct. zbMATHGoogle Scholar
  2. 2.
    Andersen S, Andersen L (2010) Modelling of landslides with the material-point method. Comput Geosci. zbMATHGoogle Scholar
  3. 3.
    ANSYS Mechanical APDL and Mechanical Applications Theory Reference, ANSYS Release 15.0, ANSYS Inc. (2010)Google Scholar
  4. 4.
    Baber TT, Wen YK (1980) Seismic response of hysteretic degrading structures, Publ. Turk. Natl. Comm. on Earthquake Eng. Volume 7, Turkish National Committee on Earthquake Engineering, Istanbul, Turkey, pp 457–464Google Scholar
  5. 5.
    Bardenhagen SG, Kober EM (2004) The generalized interpolation material point method. CMES Comput Modell Eng Sci.
  6. 6.
    Beedle LS, Christopher R (1963) Tests of steel moment connections. AISC Eng J 1(4):116–125Google Scholar
  7. 7.
    Beuth L, Wieckowski Z, Vermeer PA (2011) Solution of quasi-static large-strain problems by the material point method. Int J Numer Anal Methods Geomech.
  8. 8.
    Bouc R. (1967) Forced vibration of mechanical system with hysteresis. In: Proceedings of 4th conference on nonlinear oscillation, PragueGoogle Scholar
  9. 9.
    Brackbill JU, Ruppel HM (1986) FLIP: A method for adaptively zoned, particle-in-cell calculations of fluid flows in two dimensions. J Comput Phys. MathSciNetzbMATHGoogle Scholar
  10. 10.
    Burghardt J, Brannon R, Guilkey J (2012) A nonlocal plasticity formulation for the material point method. Comput Methods Appl Mech Eng. MathSciNetzbMATHGoogle Scholar
  11. 11.
    Casciati F (1995) Stochastic dynamics of hysteretic media. In: Krée P, Wedig W (eds) Probabilistic methods in applied physics. Lecture notes in physics, vol 451. Springer, BerlinGoogle Scholar
  12. 12.
    Charalampakis AE (2015) The response and dissipated energy of Bouc–Wen hysteretic model revisited. Arch Appl Mech.
  13. 13.
    Chen Z, Jiang S, Gan Y et al (2014) A particle-based multiscale simulation procedure within the material point method framework. Comput Particle Mech. Google Scholar
  14. 14.
    Christos SD, Vlasis KK (2016) Plane stress problems using hysteretic rigid body spring network models. Comput Particle Mech. Google Scholar
  15. 15.
    Crisfield MA (1996) Non-linear finite element analysis of solids and structures, vol 1. Wiley, New YorkGoogle Scholar
  16. 16.
    Daphalapurkar NP, Lu H, Coker D, Komanduri H (2007) Simulation of dynamic crack growth using the generalized interpolation material point (GIMP) method. Int J Fract. zbMATHGoogle Scholar
  17. 17.
    Erlicher S, Bursi OS (2008) Bouc–Wen-type models with stiffness degradation: thermodynamic analysis and applications. J Eng Mech.
  18. 18.
    Erlicher S, Point N (2004) Thermodynamic admissibility of Bouc–Wen type hysteresis models. C R Mec 332:51–57. CrossRefzbMATHGoogle Scholar
  19. 19.
    Gan Y, Sun Z, Chen Z, Zhang X, Liu Y (2017) Enhancement of the material point method using B-Spline basis functions. Int J Numer Methods Eng. Google Scholar
  20. 20.
    Harlow FH (1964) The particle-in-cell method for fluid dynamics. In: Alder B, Fernbach S, Rotenberg M (eds) Methods in computational physics, fundamental methods in hydrodynamics, vol 3. Academic Press, New YorkGoogle Scholar
  21. 21.
    Kakouris EG, Triantafyllou SP (2017) Material point method for crack propagation in anisotropic media: a phase field approach. Arch Appl Mech. Google Scholar
  22. 22.
    Liu MB, Liu GR (2010) Smoothed particle hydrodynamics (SPH): an overview and recent developments. Arch Comput Methods Eng 17:25. MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Love E, Sulsky DL (2006) An energy-consistent material-point method for dynamic finite deformation plasticity. Int J Numer Methods Eng. MathSciNetzbMATHGoogle Scholar
  24. 24.
    Ma S, Zhang X (2007) Material point method for impact and explosion problems. Comput Mech. Google Scholar
  25. 25.
    Moysidis AN, Koumousis VK (2015) Hysteretic plate finite element. J Eng Mech. Google Scholar
  26. 26.
    Nairn JA (2003) Material point method calculations with explicit cracks. Comput Model Eng Sci. zbMATHGoogle Scholar
  27. 27.
    Nairn JA, Bardenhagen SG, Smith GD (2017) Generalized contact and improved frictional heating in the material point method. Comput Particle Mech. Google Scholar
  28. 28.
    Nguyen VP, Rabczuk T, Bordas S, Duflot M (2008) Meshless methods: a review and computer implementation aspects. Math Comput Simul. MathSciNetzbMATHGoogle Scholar
  29. 29.
    Sadeghirad A, Brannon RM, Burghardt J (2011) A convected particle domain interpolation technique to extend applicability of the material point method for problems involving massive deformations. Int J Numer Methods Eng. MathSciNetzbMATHGoogle Scholar
  30. 30.
    Sadeghirad A, Brannon RM, Guilkey JE (2013) Second-order convected particle domain interpolation (CPDI2) with enrichment for weak discontinuities at material interfaces. Int J Numer Methods Eng. MathSciNetzbMATHGoogle Scholar
  31. 31.
    Steffen M, Kirby RM, Berzins M (2008) Analysis and reduction of quadrature errors in the material point method (MPM). Int J Numer Methods Eng. MathSciNetzbMATHGoogle Scholar
  32. 32.
    Steffen M, Wallstedt PC, Guilkey JE, Kirby RM, Berzins M (2008) Examination and analysis of implementation choices within the material point method (MPM). CMES Comput Model Eng Sci. Google Scholar
  33. 33.
    Sulsky D, Chen Z, Schreyer HL (1994) A particle method for history-dependent materials. Comput Methods Appl Mech Eng 118:179–196MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Sulsky D, Kaul A (2004) Implicit dynamics in the material-point method. Comput Methods Appl Mech Eng. MathSciNetzbMATHGoogle Scholar
  35. 35.
    Triantafyllou S, Koumousis VK (2012) Bouc–Wen hysteretic plane stress element. J Eng Mech.
  36. 36.
    Triantafyllou SP, Koumousis VK (2012) An hysteretic quadrilateral plane stress element. Arch Appl Mech. zbMATHGoogle Scholar
  37. 37.
    Wang B, Karuppiah V, Lu H, Komanduri R, Roy S (2005) Two-dimensional mixed mode crack simulation using the material point method. Mech Adv Mater Struct. Google Scholar
  38. 38.
    Wang B, Vardon PJ, Hicks MA, Chen Z (2016) Development of an implicit material point method for geotechnical applications. Geotech Comput. Google Scholar
  39. 39.
    Yaw LL, Kunnath SK, Sukumar N (2009) Meshfree method for inelastic frame analysis. J Struct Eng.
  40. 40.
    Zhang X, Sze KY, Ma S (2006) An explicit material point finite element method for hyper-velocity impact. Int J Numer Methods Eng. zbMATHGoogle Scholar
  41. 41.
    Zhang DZ, Ma X, Giguere PT (2011) Material point method enhanced by modified gradient of shape function. J Comput Phys. MathSciNetzbMATHGoogle Scholar

Copyright information

© OWZ 2018

Authors and Affiliations

  1. 1.Institute of Structural Analysis and Antiseismic ResearchNational Technical University of AthensAthensGreece

Personalised recommendations